Tuesday, November 24, 2009

On Buiter, Goodwin, and Nonlinear Dynamics

A few months ago, Willem Buiter published a scathing attack on modern macroeconomics in the Financial Times. While a lot of attention has been paid to the column's sharp tone and rhetorical flourishes, it also contains some specific and quite constructive comments about economic theory that deserve a close reading. One of these has to do with the limitations of linearity assumptions in models of economic dynamics:

When you linearize a model, and shock it with additive random disturbances, an unfortunate by-product is that the resulting linearised model behaves either in a very strongly stabilising fashion or in a relentlessly explosive manner. There is no ‘bounded instability’ in such models. The dynamic stochastic general equilibrium (DSGE) crowd saw that the economy had not exploded without bound in the past, and concluded from this that it made sense to rule out, in the linearized model, the explosive solution trajectories. What they were left with was something that, following an exogenous random disturbance, would return to the deterministic steady state pretty smartly. No L-shaped recessions. No processes of cumulative causation and bounded but persistent decline or expansion. Just nice V-shaped recessions.

Buiter is objecting here to a vision of the economy as a stable, self-correcting system in which fluctuations arise only in response to exogneous shocks or impulses. This has come to be called the Frisch-Slutsky approach to business cycles, and its intellectual origins date back to a memorable metaphor introduced by Knut Wicksell more than a century ago: "If you hit a wooden rocking horse with a club, the movement of the horse will be very different to that of the club" (translated and quoted in Frisch 1933). The key idea here is that irregular, erratic impulses can be transformed into fairly regular oscillations by the structure of the economy. This insight can be captured using linear models, but only if the oscillations are damped - in the absence of further shocks, there is convergence to a stable steady state. This is true no matter how large the initial impulse happens to be, because local and global stability are equivalent in linear models.

A very different approach to business cycles views fluctuations as being caused by the local instability of steady states, which leads initially to cumulative divergence away from balanced growth. Nonlinearities are then required to ensure that trajectories remain bounded. Shocks to the economy can make trajectories more erratic and unpredictable, but are not required to account for persistent fluctuations. An energetic and life-long proponent of this approach to business cycles was Richard Goodwin, who also produced one of the earliest such models in economics (Econometrica, 1951). Most of the literature in this vein has used aggregate investment functions and would not be considered properly microfounded by contemporary standards (see, for instance, Chang and Smyth 1971, Varian 1979, or Foley 1987). But endogenous bounded fluctuations can also arise in neoclassical models with overlapping generations (Benhabib and Day 1982, Grandmont 1985).

The advantage of a nonlinear approach is that it can accommodate a very broad range of phenomena. Locally stable steady states need not be globally stable, so an economy that is self-correcting in the face of small shocks may experience instability and crisis when hit by a large shock. This is Axel Leijonhufvud's corridor hypothesis, which its author has discussed in a recent column. Nonlinear models are also required to capture Hyman Minsky's financial instability hypothesis, which argues that periods of stable growth give rise to underlying behavioral changes that eventually destabilize the system. Such hypotheses cannot possibly be explored formally using linear models.

This, I think, is the point that Buiter was trying to make. It is the same point made by Goodwin in his 1951 Econometrica paper, which begins as follows:

Almost without exception economists have entertained the hypothesis of linear structural relations as a basis for cycle theory. As such it is an oversimplified special case and, for this reason, is the easiest to handle, the most readily available. Yet it is not well adapted for directing attention to the basic elements in oscillations - for these we must turn to nonlinear types. With them we are enabled to analyze a much wider range of phenomena, and in a manner at once more advanced and more elementary.

By dropping the highly restrictive assumptions of linearity we neatly escape the rather embarrassing special conclusions which follow. Thus, whether we are dealing with difference or differential equations, so long as they are linear, they either explode or die away with the consequent disappearance of the cycle or the society. One may hope to avoid this unpleasant dilemma by choosing that case (as with the frictionless pendulum) just in between. Such a way out is helpful in the classroom, but it is nothing more than a mathematical abstraction. Therefore, economists will be led, as natural scientists have been led, to seek in nonlinearities an explanation of the maintenance of oscillation. Advice to this effect, given by Professor Le Corbeiller in one of the earliest issues of this journal, has gone largely unheeded.